Foundations of the Non-Linear Mechanics of Continua

International Series of Monographs on Interdisciplinary and Advanced Topics in Science and Engineering, Volume 1: Foundations of the Non-Linear Mechanics of Continua deals with the theoretical apparatus, principal concepts, and principles used in the construction of models of material bodies that fill space continuously. This book consists of three chapters. Chapters 1 and 2 are devoted to the theory of tensors and kinematic applications, focusing on the little-known theory of non-linear tensor functions. The laws of dynamics and thermodynamics are covered in Chapter 3.

This volume is suitable for persons who intend to do research on the development of the theory of dynamics and thermodynamics or solve specific theoretical problems on the motion of a continuous medium with finite deformations.

Editor's Preface

Foreword

Introduction

Chapter I. Outline of Tensor Calculus

1. Curvilinear Systems of Coordinates

2. Transformation of Coordinates

3. The Concept of a Tensor and the Fundamentals of the Algebra of Tensors

4. Tensors of Second Rank

5. Tensor Functions

6. Functions of Several Tensors

7. Tensor Potential Functions

8. Differentiation of Tensors with Respect to Space Coordinates

9. The Riemann-Christoffel Tensor

10. Differentiation of Tensors with Respect to Parameters

Chapter II. Kinematics of a Deformable Medium

1. General Features of Continuous Deformations

2. Affine Deformations

3. Regular Finite Displacements of a Continuum

4. The Deformation Tensors

5. The Vector of Rotation of the Axes of Deformation

6. Kinematic Tensors Characterizing a Deformation

7. The Differentiation of Non-Linear Tensor Functions

8. The Conditions of Compatibility

Chapter III. Dynamic and Thermodynamic Exposition

1. Physical Considerations

2. The Concept of a Material Continuum

3. The Equations of Continuity

4. Dynamical Equations

5. The Theorem of Kinetic Energy and the Work of the Internal Surface Forces

6. Thermodynamic Systems and Cycles. The Equation of Conservation of Energy and the Concept of the Internal Energy of a System

7. The Basic Concepts and Consequences of the Second Law of Thermodynamics

8. Ideal Liquids and Gases

9. The Simplest Ideal Processes

10. Special Examples of Ideal Compressible Media

11. The Equations of Motion of Gas Mixtures with Physico-Chemical Interaction Between the Components

12. Viscous Liquids and Gases

13. The Turbulent Flow of Continuous Media

14. The Model of an Elastic Body

15. Thermoelastic Isotropic Media

16. Some Remarks on Mechanical Models for Irreversible Processes

17. Plastic, Elastic and Total Deformations

18. The Elastic Range and the Loading Surface

19. The Basic Laws in the Theories of Plastic Bodies

20. Models of Plastic Media for which the Loading Surface Possesses Singular Points

References

Index